I know that in an elimination style tournament, a player who gets a red card must sit out in the next game. In a round robin tournament such as the Premier League, what is the consequence of the red card in future games?
Red card means suspension, however how long the player gets suspended might vary. If the player was shown a second yellow card, and thereby a red card. He typically misses the next game.
On the other hand if the player gets a direct red it's often a longer suspension. I recall something about a two-match default suspension (against which you can appeal) and a referral to the discipline committee, for possibly longer suspension. But I can't find a clear cut reference about it. At any rate the following is what the English Football Association has about suspensions:
The two categories of match are first team competitive matches and non-first team matches. Premier League, Football League, Conference National, WSL:
Red cards in first team competitive matches result in immediate suspension from forthcoming first team competitive matches. Red cards in non-first team matches will take effect from the 14th day following dismissal.
Clubs at Step 2 and below: Red cards in first team competitive matches result in suspensions commencing on the 14th day following the match in which the player was sent off. Red cards in non-first team matches will be processed by the County Association to which the Club is affiliated.
The rule is the same. It's actually possible to be suspended in a round-robin tournament for a red card in a direct elimination match or vice versa. Suspensions also carry over from one season to the next.
For example, a red card in a FIFA World Cup final would result in a suspension that is carried over to the team's next official match. A red card in a championship match would result in a suspension for the club's next official match (including cup's matches).
For clubs, cautions and suspensions are tallied separately for national league or cup matches on the one hand and UEFA matches on the other hand. Friendly matches are also treated separately.